Holographic perfect uidity, Cotton energy-momentum duality ...

Introduction Stationary fluids Fluids and holography Exact bulk reconstruction Conclusions

Holographic perfect fluidity, Cotton

energy-momentum duality and transport

properties

Valentina Pozzoli

CPHT - Ecole Polytechnique

based on arXiv:1309.2310 and arXiv:1206.4351

with M. Caldarelli, R. Leigh, A. Mukhopadhyay,

A. Petkou, M. Petropoulos, K. Siampos

Holographic perfect fluidity, Cotton energy-momentum duality and transport properties Valentina Pozzoli CPHT - Ecole Polytechnique


Outline

1 Introduction

2 Stationary fluidsFluids on curved backgroundsRanders–Papapetrou geometries

3 Fluids and holographyThe holographic expansionSome examples

4 Exact bulk reconstructionMonopolar geometriesDipolar geometriesExact bulk metric

5 Conclusions



The fluid/gravity correspondence

Fluid/gravity correspondence: limit of AdS/CFT when the boundary iswell approximated by its long wavelength description.

Stationary black holesin 3+1 dimensions

2+1-dimensional perfect fluidswith non-trivial vorticity

Motivations

• Applications in AdS/CMT (Bose fast-rotating gases, ... )

• Analogue gravity interpretation (applications to meta-materials)

• Search of new black hole solutions

• Information on transport coefficients



Outline

1 Introduction




5 Conclusions



Fluids on curved backgrounds

Fluid dynamics on 2+1 dimensional curved backgrounds

Tµν function of uµ, gµν , and their covariant derivatives satisfies Eulerequations

∇µTµν = 0.

The energy–momentum tensor can be expanded as

Tµν = Tµν(0) + Tµν

(1) + Tµν(2) + · · · ,

where the subscript denotes the number of covariant derivatives and

Tµν(0) = εuµuν + p∆µν , ∆µν = uµuν + gµν .

→ Perfect-fluid energy–momentum tensor.→ Higher-order corrections involve the presence of transport coefficients.




Transport coefficients

Higher order corrections:

Tµν(1) = −

(2ησµν + ζ∆µνΘ + ζHε

ρλ(µuρσλν)),

aµ = uν∆νuµ acceleration,σµν = ∇(µuν) + a(µuν) − 1

2 ∆µν∇ρuρ shear,

Θ = ∇µuµ expansion.

Transport coefficients

η, ζ, ζH , . . . transport coefficients.

• they give information on the microscopic theory,

• dissipative and non-dissipative.




Conditions for perfect equilibrium

If we want the fluid on a curved background to be in perfect equilibrium

→ the transport coefficient is vanishing,

→ the corresponding tensor is vanishing.

For example, in Minkowskian backgrounds any inertial fluid hasTµν = T perf

µν .

→ Depending on the geometry, different sets of transport coefficients arerequired to vanish.



Randers–Papapetrou geometries

Randers–Papapetrou backgrounds

We consider stationary metrics with unique normalized time-like Killingvector:

ds2 = −(dt − bidxi )2 + aijdx

idx j .

Velocity one-form u = −dt + bidxi with vorcity ω = 1

2db.

In 2+1 dimensions

- Vorticity ωµν = − q2ηµνρu

ρ, where q(x) is a scalar field.

- Curvature Rµν .

- Cotton–York tensor Cµν = εµρσ∇ρ(Rνσ − 1

4Rδνσ).



Outline

1 Introduction




5 Conclusions



The holographic expansion

The holographic expansion

Bulk solutions of general relativity with Λ = −3k2 can always be taken inthe form

ds2 = ηabθaθb = dr2

k2r2 + k2r2ηµνθµθν ≈ dr2

k2r2 + k2r2g(0)µνdxµdxν

→ The fluid boundary data is read at r →∞.

Expansion for large r of the bulk orthonormal frame:

θµ(r , x) = krEµ(x) + 1kr F

µ[2](x) + 1

k2r2 Fµ(x) + · · ·

→ only two independent data:

- the boundary metric ds2 = ηµνEµEν = g(0)µνdxµdxν

- the boundary energy–momentum tensor T = κFµeµ = TµνE

ν ⊗ eµ.



Some examples

Schwarzschild black hole

Bulk data

ds2 = dr2

V (r) − V (r)dt2 + r2(dθ2 + sin2 θdφ2

),

V (r) = 1 + k2r2 − 2M/r .Holographic coordinate: θr = dr/

√V (r) = dr/kr .

Boundary data

Boundary metric: ds2bdy = −dt2 + 1

k2

(dθ2 + sin2 θdφ2

).

Boundary energy–momentum tensor:T = κMk

3

(2dt2 + 1

k2

(dθ2 + sin2 θdφ2

)).

→ static perfect fluid with no vorticity.



Some examples

Taub-NUT black hole

Bulk data

ds2 = dr2

V (r) − V (r) (dt − 2ν cos θdφ)2 + ρ2(dθ2 + sin2 θdφ2

),

V (ρ) = ∆r/ρ2 with

∆r = (r2 − ν2)(1 + k2(r2 + 3ν2)) + 4k2ν r2 − 2Mr ,ρ2 = r2 + ν2.

Boundary data

Boundary metric:ds2

bdy = − (dt + 2ν(1− cos θ)dφ)2 + 1k2

(dθ2 + sin2 θdφ2

).

Velocity: u = −dt + bidxi = −dt + 2ν(cos θ − 1)dφ.


3

(2u2 + 1

k2

(dθ2 + sin2 θdφ2

)).



Some examples

Taub-NUT boundary

→ Perfect fluid with vorticity:

ω = 12 db = −ν sin θdθ ∧ dφ.

In the bulk: non-rigid rotation with angular momentum distributionalong the Misner string.In the boundary: monopolar-like vorticity.

• North pole: angular velocity Ω∞ = νk2.

• South pole: no angular velocity.



Some examples

Kerr black hole

Bulk data

ds2 = dr2

V (r) − V (r)(dt − a

Ξ sin2 θdφ)2

+ ρ2

∆θdθ2 + sin2 θ∆θ

ρ2

(a dt − r2+a2

Ξ dφ)2

,

V (ρ) = ∆r/ρ2 with

∆r = (r2 + a2)(1 + k2r2)− 2Mr , ρ2 = r2 + a2 cos2 θ,∆θ = 1− k2a2 cos2 θ, Ξ = 1− k2a2.

Boundary data

Boundary metric:

ds2bdy = −

(dt − a sin2 θ

Ξ dφ)2

+ 1k2∆θ

(dθ2 +

(∆θ sin θ

Ξ

)2dφ2).

Velocity: u = −dt + bidxi = −dt + a sin2 θ

Ξ dφ.


3

(2u2 + 1

k2

(dθ2 + sin2 θdφ2

)).



Some examples

Kerr boundary

→ Perfect fluid with vorticity:

ω = 12 db = a cos θ sin θ

Ξ dθ ∧ dφ.

In the bulk: rigid rotation.In the boundary: dipolar-like vorticity.

Constant boundary angular velocity Ω = −ak2.



Outline

1 Introduction




5 Conclusions



Exact bulk reconstruction

Aim: find boundary geometries such that

g(0)µν , T perfµν

exact 3+1 dimensionalEinstein geometry

Answer: perfect-Cotton geometries Cµν = cT perfµν , with c constant:

Cµνdxµdxν = c

2

(2u2 + d`2

).

Valid in general, but explicit solution when an extra isometry is present:

• Monopolar geometries.

• Dipolar geometries.



Monopolar geometries

Monopolar geometries

→ The vorticity is constant.

Boundary properties

• The boundary is an homogeneous space → the space has thestructure of fibrations over S2,R2 or H2.

• No possible correction to the energy–momentum tensor can be built→ there are no constraint on the transport coefficients.

Bulk properties

Bulk geometries: Taub-NUT black holes with regular horizons.



Dipolar geometries

Dipolar geometries

→ The vorticity is not constant, but the space is conformally flat.

Boundary properties

• Axisymmetric spaces: global rigid rotation.

• The geometry allows for corrections to the energy–momentumtensor → holography puts constraints on an infinite number oftransport coefficients.

Bulk properties

Bulk geometries: Kerr black holes with regular horizons.

It is possible to consider general monopolar-dipolar geometries: thecorresponding bulk is Kerr-Taub-NUT.



Exact bulk metric

Solutions with one isometry

Randers–Papapetrou boundary

ds2bd = −u2 + d`2, R = R + q2

2

Bulk uplift of a perfect–Cotton geometry:

ds2 = −2u

(dr − 1

2dxρuσηρσµ

∇µq

)+ ρ2d`2 −

(r2 +

δ

2− q2

4− 1

ρ2(2Mr +

qc

2)

)u2,

δ = R + 3q2, ρ2 = r2 + q2

4 .

- It is a solution of Einstein’s equations.

- It contains the Kerr-Taub-NUT black holes as well as other solutions.



Outline

1 Introduction




5 Conclusions



Conclusions

Holographic fluids

Sufficient conditions for the correspondence to be exact and holographicconstraint of the transport coefficients.

Further directions

• Holography as a bottom-up solution generating technique.

• Probe more transport coefficients: higher multipole and perturbativeapproaches.


Holographic perfect uidity, Cotton energy-momentum duality ...

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